A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves, as distinct from a small circle. The great circle therefore has both the same circumference and the same center as the sphere. It is the largest circle that can be drawn on a given sphere.
Great circles serve as the analogue of "straight lines" in spherical geometry. See also spherical trigonometry and geodesic.
The great circle, also known as the Riemannian circle, is the path with the smallest curvature, and hence, an arc (or an orthodrome) of a great circle is the shortest path between two points on the surface. The distance between any two points on a sphere is therefore known as the greatcircle distance. The greatcircle route is the shortest path between two points across the surface of a sphere
Strictly speaking the Earth is not a perfect sphere (it's an oblate spheroid or ellipsoid  i.e., slightly squashed at the poles), which means that the shortest distance between two points (a geodesic) is not quite a great circle. Nevertheless, the sphere model can be considered a first approximation.
When long distance aviation or nautical routes are drawn on a flat map (for instance, the Mercator projection), they often look curved. This is because they lie on great circles. A route that would look like a straight line on the map would actually be longer. An exception is the gnomonic projection, in which all straight lines represent great circles.
On the Earth, the meridians are on great circles, and the equator is a great circle. Other lines of latitude are not great circles, because they are smaller than the equator; their centers are not at the center of the Earth  they are small circles instead. Great circles on Earth are roughly 40,000 km in length, though the Earth is not a perfect sphere; for instance, the equator is 40,075 km.
Some examples of great circles on the celestial sphere include the celestial horizon, the celestial equator, and the ecliptic.
Great circle routes are used by ships and aircraft where currents and winds are not a significant factor. Flight lengths can therefore often be approximated to the greatcircle distance between two airports. For aircraft travelling west between continents in the northern hemisphere these paths will extend northward near or into the Arctic region, however easterly flights will often fly a more southerly track to take advantage of the jet stream.
If one were to travel along a great circle, it would be difficult to steer manually as the heading would constantly be changing (except in the case of due north, south, or along the equator). Thus, Great Circle routes are often broken into a series of shorter rhumb lines which allow the use of constant headings between waypoints along the Great Circle.
]] A great circle the largest possible circle that can be drawn on a sphere, one that divides the surface into equal halves, called hemispheres. It is a circle that has the same diameter as the sphere it was drawn on. These curves are geodesics in the sphere and all have the same circumference, that is, the length of the circle.
There are an infinite number of great circles that can be drawn on any perfect sphere. The latitude lines on a globe all form great circles that pass through the same two points (the North Pole and the South Pole). The Equator is another great circle.
Great Circles can be used to determine the shortest surface distance between two points on a sphere (or on the earth).
A straight line from plane Euclidean geometry corresponds to a Great Circle in nonEuclidean spherical geometry.
